Q$1.$ Tea exports $($thousands of tons$)$ as well as interest rates were recorded for the years as indicated in the table below.

Year $123456789101112131415$

Int. Rt $5812141517141821222023242521$

Tons $978183767863726254585150454752$

If the volume of tea exported is thought to regress on interest rates,

$($a$)$ Test for heteroscedasticity by Spearman$$s rank correlation test.

$($b$)$ Carry out a Durbin$-$Watson auto correlation test.

Q$2$ In a study of the relationship between ownership of a farm tractor $($Y$)$ and maize harvest $($X$)$ on maize farmers in a region, a sample of farmers whose volume in harvest is was randomly selected and the number of farmers owning tractors recorded. If is the sample size from the group with volume and is the number in the group with tractors as in the table below,

xi $212630131520253035404447$

Ni $657754576470554030233328$

ni $122418222536352725192823$

$($a$)$ Fit a logit model and estimate the ownership of a tractor for a farmer whose harvest is $18.$

$($b$)$ Fit a probit model and estimate the ownership of a tractor for a farmer whose harvest is $18.$